Ibrahim Assem

The fundamental groups of a triangular algebra

A finite dimensional algebra A (over an algebraically closed field) is called triangular if its ordinary quiver has no oriented cycles. To each presentation (Q I) of A is attached a fundamental group π1(Q I), and A is called simply connected if π1(Q I) is trivial for every presentation of A. In this paper, we provide tools for computations with the fundamental groups, as well as criteria for simple connectedness. We find relations between the fundamental groups of A and the first Hochschild cohomology H 1 (A A).

Minimal representation-infinite coil algebras

For a basic and connected finite dimensional algebra A over an algebraically closed field, we study when the cycles in the category mod A (of finite dimensional modules) are well-behaved. We call A cycle-finite if, for any cycle in mod A, no morphism on the cycle lies in the infinite power of the radical. We show that, in this case, A is tame. We also introduce a natural generalisation of a tube, called a coil, and define A to be a coil algebra if any cycle in mod A lies in a standard coil. We prove that the minimal representation-infinite coil algebras coincide with the tame concealed algebras.

Two-sided gluings of tilted algebras

We study the class of algebras A satisfying the property: all but at most finitely many non-isomorphic indecomposable A-modules are such that all their predecessors have projective dimension at most one, or all their successors have injective dimension at most one. Such a class includes the tilted algebras [D. Happel, C. Ringel, Trans. Amer. Math. Soc. 274 (1982) 399–443], the quasi-tilted algebras [D. Happel, I. Reiten, S. Smalo, Mem. Am. Math. Soc. 120 (1996) 575], the shod algebras [F.U. Coelho, M. Lanzilotta, Manuscripta Mathematica 100 (1999) 1–11], the weakly shod [F.U. Coelho, M. Lanzilotta, Preprint, 2001], and the left and right glued algebras [I. Assem, F.U. Coelho, J. Pure Appl. Algebra 96 (3) (1994) 225–243].

Glueings of tilted algebras

Abstract Let A be a basic connected finite-dimensional algebra over an algebraically closed field. We show that if A has all its indecomposable projectives (or injectives) lying in a component of the Auslander—Reiten quiver consisting entirely of postprojective (or preinjective, respectively) modules in the sense of Auslander and Smalo then A is a finite enlargement in the postprojective (or preinjective, respectively) components of a finite set of tilted algebras having complete slices in these components. We call such an algebra A a left (or right, respectively) glued algebra and study some of its homological properties in particular in the case where A is itself a tilted algebra.

Representation-finite trivial extension algebras

Abstract Let A be a finite-dimensional basic connected associative algebra over an algebraically closed field, and T(A)=A ⋉ DA its trivial extension by its minimal injective cogenerator. We prove that T(A) is representation-finite of Cartan class Δ if and only if A is an iterated tilted algebra of Dynkin class Δ. The proof also yields a construction procedure for iterated tilted algebras of Dynkin type.

Tilting modules over split-by-nilpotent extensions

Let a finite dimensional algebra R be a split extension of an algebra A by a nilpotent bimodule Q. We give necessary and sufficient conditions for a (partial) tilting module TA to be such that T⊗A RR is a (partial) tilting module. If this is not the case, but QA is generated by the tilting module TA , then there exists a quotient [Rbar] of R such that T⊗A [Rbar][Rbar] is a tilting module.

Quadratic forms and iterated tilted algebras

Given a finite quiver A without oriented cycles, an algebra A is called an iterated tilted algebra of type A [2] if there exists a sequence of algebras A = A,, A 1) . ..) A,,,, where A, is the path algebra of A, and a sequence of tilting modules T>, (0 , and every indecomposable A ,-module satisfies either Hom,J T’, M) = 0 or Exta,( T’, M) = 0. If m < 1, A is called a tilted algebra of type A [25]. The representation theory of iterated tilted algebras was proved to be closely related to that of a class of symmetric algebras, namely, the trivial extension algebras; see [3,4, 18, 27, 291. Recently, they were also shown to arise naturally in the study of the derived category of a finite dimensional algebra; see [23, 24, 71. Iterated tilted algebras of type A, where the underlying graph of A is a Dynkin diagram, were studied in [ 1, 2, 8, 223, and the iterated tilted algebras of Euclidean type A,,, (m 3 1) were classified in [S]. 55 0021-8693/90

FRIEZES, STRINGS AND CLUSTER VARIABLES

3.00

Laura Skew Group Algebras

To any walk in a quiver, we associate a Laurent polynomial. When the walk is the string of a string module over a 2-Calabi-Yau tilted algebra, we prove that this Laurent polynomial coincides with the corresponding cluster character of the string module, up to an explicit normalising monomial factor.

ENDOMORPHISM ALGEBRAS OF PROJECTIVE MODULES OVER LAURA ALGEBRAS

We prove that if A is an artin algebra, G is a finite group acting on A such that |G| is invertible in A, and R = A[G]b is a basic algebra associated with the skew group algebra, then A is left supported (or right supported, or laura, or left glued, or right glued, or weakly shod, or shod) if and only if so is R.